User blog:Tetramur/My thoughts about functions and numbers
I thought about various googological functions and numbers. That's my version of extending googology past computability. Fast-growing hierarchy \(f_0(n) = n + 1\) \(f_{\alpha+1}(n)\) = \(f_{\alpha}^n(n)\) \(f_{\alpha}(n)\) = \(f_{\alphan}(n)\) if \(\alpha\) is a limit ordinal and \(\alpha n\) is a n-th member of fundamental sequence. For small ordinals it is not difficult to define a FS. However, how will you define a FS for, say, \(\omega^\text{CK}_1\)? There is a Kleene's O notation, but how do you distinguish pathological and non-pathological enumeration of TMs? So, any currently known definition of \(f_{\omega^\text{CK}_1}(n)\) is not consistent, therefore it is ill-defined by now. Any part of FGH for higher ordinals is also ill-defined. But in this work, I suppose that some non-pathological enumeration exists and is given, and this part of FGH is well-defined. If you got this, we can continue. Busy Beaver, TMs, ITTM etc. It is known that there are some functions which are represented in FGH with countable ordinal, but they are uncomputable. For example, take some enumeration of TMs and give value 1 to the function if TM #n halts and 0 if it doesn't halt. This function is uncomputable, but it is nowhere within the range of fast-growing functions! So, my previous definition of weakly/strongly uncomputable functions must be revised. It was as follows: 1. The weakly uncomputable function is the function which can be represented with countable non-recursive ordinal in FGH if the appropriate fundamental sequences (hereafter FS) are given. 2. The strongly uncomputable function is the function which can be represented with uncountable ordinal in FGH if the appropriate FS are given. Now, it would be as follows: 1. The weakly uncomputable function of order n-1 is the function which can be computed with TM + oracle of order n, but not lower oracle. 2. The strongly uncomputable function is the function which has a relationship with a set theory or its extensions. It is generally supposed that ordinary busy beaver's function grows approximately as \(f_{\omega^\text{CK}_1}(n)\) in the FGH associated to Kleene's \(\mathcal{O}\). Now, oracle order-k function. That is, analogue of busy beaver but with an oracle. I suppose that the oracle order-k function is growing as \(f_{\omega^\text{CK}_{k+1}}(n)\) - again, if the appropriate FS are given. Why? \(\omega^\text{CK}_1\) is a supremum of all ordinals which correspond to all computable functions for ordinary TMs. I think \(\omega^\text{CK}_n\) would be a supremum of all ordinals which correspond to all (hyper-)computable functions for TMs with an oracle of order n-1. For this, I think that analogue of BB with an oracle of order n would grow as \(f_{\omega^\text{CK}_{k+1}}(n)\). So, I suppose the Fish Number 4 is \(f_{\omega^\text{CK}_{(\omega^{\omega+1}) 63}}(63)\) if the appropriate FS are given. ... Rayo's function and its extensions Today I have two main conjectures about Rayo's function: 1. Rayo's function would eventually dominate every function represented in FGH with countable ordinal (doesn't matter whether this function is computable or uncomputable) if it were properly formalized and well-defined, and 2. Rayo's function would be on the level of \(f_{\omega_1}(n)\) if the appropriate FS were given and if it were properly formalized and well-defined. But do they exist? Really, no, they don't exist. Any existence of FS for \(\omega_1\) would imply a contradiction. ... (page will be finished, but it will be not soon) Category:Blog posts